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Children soon learn that they can pump up the swing amplitude by lowering their center of gravity on the downswing and raising it on the upswing They also learn that this pumping must be done at twice the swing resonance frequency When they find this out, you do not need to push their swing any more: They have realized a sustained oscillator But the sort of sustained oscillator that led to the maser and laser is different Consider a typical La oscillator It was the Irish physicist George Francis FitzGerald who first suggested using this oscillator to generate radio waves [213] His idea was later adopted by Hertz in his famous experimental confirmation of Maxwell's theory of electrodynamics As any actual circuit always has some resistance, the oscillations eventually die out.



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The use of a strong base alone for lactam polymerization is limiting. There are the induction periods previously noted and, more importantly, only the more reactive lactams, such as E-caprolactam and 7-heptanolactam ( -enantholactam), readily undergo polymerization. The less reactive lactams, 2-pyrrolidinone and 2-piperidinone, are much more sluggish toward polymerization by strong base alone. Formation of the imide dimer is dif cult from these relatively unreactive lactams. Both limitations are overcome by forming an imide by reaction of lactam with an acylating agent such as acid chloride or anhydride, isocyanate, monocarbodiimide, and others [Luisier et al., 2002]. Thus, E-caprolactam can be readily converted to an N-acylcaprolactam XXXIX by reaction with an acid chloride. The N-acyllactam can be synthesized in situ or preformed and then added to the reaction system.

7-64





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To make a sustained oscillator out of a good old La circuit, what we need is to couple it to some amplifying element to effectively cancel the resistance in the circuit In the early years of radio, this amplifying element was the negative resistance voltaic arc [460] The first mathematical analysis of this sustained La oscillator, however, was carried out by van der Pol for a vacuum tube, which soon replaced the arc as a practical amplifying element Figure 81 shows the diagram of the circuit considered by van der Pol He derived the following differential equation for the deviation v from the mean of the anode voltage:.

Initiation consists of the reaction of the N-acyllactam with activated monomer followed by fast proton exchange with monomer

3) +wov = 0, 2

7-65

(8.1)

7-66

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where Wo = 1/vTC, a is the linear gain, and 11 is the saturation coefficient. 3 In the case where Wo a,11, it is useful to write v as the product of a slowly varying amplitude V(t) and a rapidly oscillating Wo frequency component (see Sec. 4-1 of [533]): v(t) = V(t) e- iwot + V*(t)e iwot . (8.2) Then it is a good approximation to neglect the second derivative of V and the products onl and a and 11, when (8.2) is substituted in (8.1). This is the slowly varying envelope approximation. Moreover, the effect on, say, the V exp( - iwot) component of v from

Species XL and XLI correspond to species XXXV and XXXVI for polymerization in the absence of an acylating agent. The acylating agent achieves facile polymerization of many

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der Pol [6091.

lactams by substituting the fast initiation sequence in Eqs. 7-65 and 7-66 for the slower sequence in Eqs. 7-60 and 7-61. The use of an acylating agent is advantageous even for the more reactive lactams as induction periods are absent, polymerization rates are higher, and lower reaction temperatures can be used. Polymerizations in the absence and presence of an acylating agent are often referred to as nonassisted and assisted polymerizations, respectively. The terms nonactivated and activated polymerization are also used, but this is confusing since both reactions involve the lactam anion, which is usually referred to as activated monomer. The literature is also confusing with regard to terms used to describe the acylating agent. Activator, promoter, catalyst, and initiator are terms that have been used. Terminology for the strong base is also a problem since its role is not the same for activated and nonactivated polymerizations. The base is needed in both polymerizations for forming the lactam anion (activated monomer). For nonassisted polymerization, the lactam anion is not only the real monomer but is also the species required to form the initiating species (imide dimer). This text will avoid the use of any of these terms for the strong base and acylating agent. Propagation follows in the same manner as for propagation of species XXXVII except that the propagating chain has an acylated end group instead of the amine end group:

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